Analog Allpass Filters
It turns out that analog allpass filters are considerably simpler mathematically than digital allpass filters (discussed in §B.2). In fact, when working with digital allpass filters, it can be fruitful to convert to the analog case using the bilinear transform (§I.3.1), so that the filter may be manipulated in the analog plane rather than the digital plane. The analog case is simpler because analog allpass filters may be described as having a zero at for every pole at , while digital allpass filters must have a zero at for every pole at . In particular, the transfer function of every firstorder analog allpass filter can be written as
This simplified rule works because every complex pole is accompanied by its conjugate for some .
Multiplying out the terms in Eq.(E.14), we find that the numerator polynomial is simply related to the denominator polynomial :
As an example of the greater simplicity of analog allpass filters relative to the discretetime case, the graphical method for computing phase response from poles and zeros (§8.3) gives immediately that the phase response of every real analog allpass filter is equal to twice the phase response of its numerator (plus when the frequency response is negative at dc). This is because the angle of a vector from a pole at to the point along the frequency axis is minus the angle of the vector from a zero at to the point .
Lossless Analog Filters
As discussed in §B.2, the an allpass filter can be defined as any filter that preserves signal energy for every input signal . In the continuoustime case, this means
where denotes the Dirac ``delta function'' or continuous impulse function (§E.4.3). Thus, the allpass condition becomes
Suppose is a rational analog filter, so that
(We have normalized so that is monic () without loss of generality.) Equation (E.15) implies
If , then the allpass condition reduces to , which implies and , in which case for all .

and , i.e.,
By analytic continuation, we have
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